Transpose Of A Matrix

If A = [aij] be a matrix of order m x n, then the matrix obtained by interchanging the rows and columns of A is known as the transpose of A. It is represented by AT.

Hence if A = [aij]mxn, then AT = [aji]mxn

Example: If A =, then AT =

PROPERTIES OF TRANSPOSE OF THE MATRICES:

If A and B be two matrices of any suitable order then,

(AT)T = A

(A ± B)T = AT ± BT

(kA)T = kAT

(AB)T = BTAT

The above properties can be verified by examples which are given below:

Example:

Find the transpose of the matrix
A = and verify that (AT)T = A.

Solution:

By interchanging the rows and columns of the matrix A we get the transpose of matrix A. Hence transpose of
matrix A = AT =
.

Now (AT)T = = A

Hence, (AT)T = A

Example: If A = , verify that (A ± B)T = AT ± BT.

Solution:

Thus, (A + B)T = AT + BT

Example: If A = ,
then verify that (kA)T = kAT where k is any constant.

Solution:

Given that A = , then

Then

Hence (kA)T = kAT

Example: If
and , verify that (AB)T = BTAT.

Solution:

Hence

BTAT =

Hence (AB)T = BTAT

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